CN109118553A - Electrical impedance tomography content Boundary Reconstruction method based on geometric constraints - Google Patents
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Abstract
The electrical impedance tomography content Boundary Reconstruction method based on geometric constraints that the present invention relates to a kind of, comprising: 1) using the boundary of part arc length parameters x (s) characterization target contents；2) energy function based on residual error function and geometric constraints building shape inversion problem: 3) according to variation principle, the Optimal Boundary estimation x for minimizing energy function ε (x) meets Lagrange's equation；Sliding-model control is carried out to the variable in energy function, content boundary is characterized by series of discrete point [x (s1), x (s2) ..., x (sN)]；4) Lagrange's equation is iteratively solved using semi-implicit method；5) by the sampled point in successive ignition back boundary estimated value can Step wise approximation target contents real border, realize content Boundary Reconstruction.
Description
Technical field
The invention belongs to electrical impedance tomography technical fields, are related to a kind of content boundary based on geometric constraints
Method for reconstructing.
Background technique
Electrical impedance tomography (Electrical Impedance Tomography, abbreviation EIT) is that one kind has non-invade
Enter or the process visualization on-line monitoring technique of non-disturbance feature.It by being placed in the array-type sensor of sensitivity field to be measured,
Apply electrical stimuli signal to target object field, and the electrical response letter of reflection sensitivity field internal conductance rate distributed intelligence can be obtained
Number, and then realize the two-dimensional/three-dimensional visualization of dielectric distribution in field.The technology has portable, inexpensive and high time resolution
The advantages that, there is wide application value in industry and biomedical aspect.However, the image reconstruction problem of EIT has seriously
Non-linear and pathosis, this causes the spatial resolution of EIT lower, and vulnerable to noise jamming influence.For nonlinear solution
Certainly method is usually to use iterative linearized or direct non-linear method.And pathosis can then pass through the benefit to prior information
With being improved, common the way of restraint has smoothness constraint, sparse constraint etc..From other measurement patterns, such as ultrasonic measurement, calculate
Hydrodynamics, dynamic data sequence etc., the prior information of acquisition also contribute to being promoted the spatial resolution of EIT.
EIT can be applied to the measurement of the bubble in such as Diagnosis of Breast Tumor, long-term monitoring of respiration and fluid field,
The distribution of conductivity observed in domain in these applications is approximately piecewise constant, therefore the target of EIT is to rebuild limited be embedded in
Simply connected subregion in homogeneous background conductivity, the content that these simply connected subregions are rebuild needed for being exactly.This segmentation
Constrainted constants have the advantage for retaining content shape and enhancing content boundary resolution, therefore have attracted the field EIT
Many concerns.
2007, S.Babaeizadeh et al. was published in " IEEE Transactions OnMedical Imaging
(IEEE medical image processing) " volume 26, the 637-647 pages, entitled " Electrical impedance tomography
for piecewise constant domains using boundary element shape-based inverse
Solutions (the EIT Piecewise Constant number field inverse problem solution based on boundary element shape) ", it proposes a kind of using Piecewise Constant
The content Boundary Reconstruction method of number constraint, this method parameterize the boundary of target contents, Zhi Houtong by one group of form factor
It crosses minimum shape energy function and optimal content boundary estimation is calculated.It, should due to need to only rebuild the boundary of content
Method is also referred to as the method for reconstructing based on shape, it reduces one-dimensional freedom degree compared to method for reconstructing pixel-based, because
This content Boundary Reconstruction method holds out broad prospects in terms of the pathosis for improving EIT, the spatial resolution for promoting EIT.It is typical
Boundary Reconstruction method there is H.Haddar in 2014 et al. to be published in " Complex Variables and Elliptic
Equations (complex function and elliptic equation) " volume 59, the 863-882 pages, entitled " A conformal mapping
Method in inverse obstacle scattering (the conformal transformation method in obstacle backscattering) " propose it is conformal
Converter technique, F.Cakoni in 2012 et al. are published in " Inverse Problems (inverse problem) " volume 29,015005-
Page 015027, entitled " Integral equation methods for the inverse obstacle problem with
The generalized impedance boundary condition (integral equation of the inverse problem under Generalized impedance boundary conditions
Method) " integration method that proposes, and certain methods based on boundary element method.
In the design of content Boundary Reconstruction method, key factor is the improvement to boundary parameterized model.General
Boundary parameter model is divided into two kinds, global boundary model and local boundary model.Global boundary model, such as: 2007
S.Babaeizadeh et al. is published in that " IEEE Transactions On Medical Imaging is (at IEEE medical image
Reason) " volume 26, the 637-647 pages, entitled " Electrical impedance tomography for piecewise
Constant domains using boundary element shape-based inverse solutions (is based on side
The EIT Piecewise Constant number field inverse problem solution of boundary's member shape) " the spherical harmonics model and D.K.Han in 1999 etc. that use
People is published in " Journal ofComputational Physics (computational physics magazine) " volume 155, the 75-95 pages, inscribes
For " A shape decomposition technique in electrical impedance tomography (electrical impedance
Shape decomposition technology in tomography) " use Fourier model, the overall situation of object boundary can be embodied in reconstruction process
Geometrical property, such as flatness.And local boundary model is usually defined in if some local geometric characteristics such as curvature etc., for example,
S.Babaeizadeh in 2007 et al. is published in " IEEE Transactions On Medical Imaging (IEEE medicine shadow
As processing) " volume 26, the 637-647 pages, entitled " Electrical impedance tomography for piecewise
Constant domains using boundary element shape-based inverse solutions (is based on side
The EIT Piecewise Constant number field inverse problem solution of boundary's member shape) " the B- Spline Model and M.M.Zhang in 2017 etc. that use
People is published in " IEEE Sensors Journal (IEEE sensor magazine) " volume 17, the 8263-8270 pages, entitled
《Quantitative reconstruction ofthe exteriorboundary shape of metallic
Inclusions using electrical capacitance tomography is (using capacitance chromatography imaging Quantitative Reconstruction gold
Belong to field trash outer boundary shape) " use the model built on divergent boundary point.Two kinds of boundary models have respective excellent
Gesture and limitation.On the one hand, since global boundary model contains the process of regularization, the shape based on global boundary model
Method for reconstructing is more stable than the method based on local boundary model.On the other hand, since local boundary model is with higher
Local deformation freedom degree, therefore local boundary model is more more flexible than global boundary model.For the shape of some complexity
Such as concave shape can be more effectively carried out characterization using local boundary model.
Summary of the invention
For the present invention in content Boundary Reconstruction method, global and local parametrization boundary model cannot meet height simultaneously
The problem of Boundary Reconstruction of stability and pinpoint accuracy, proposes a kind of content Boundary Reconstruction side based on geometric constraints
Method, this method carries out regularization constraint to local boundary parameter model using geometric constraints, and devises new energy
It minimizes equation and is used to restrained boundary Problems of Reconstruction, to reach the pathosis of improvement electrical impedance tomography problem while improve
The purpose of Boundary Reconstruction stability and precision.Technical solution is as follows:
A kind of electrical impedance tomography content Boundary Reconstruction method based on geometric constraints, including the following steps:
1) using the boundary of part arc length parameters x (s) characterization target contents, wherein x ∈ Γ is indicated on object boundary
Point set, s ∈ [0,1] are local arc length parameters, construct sensitivity matrix J and obtain boundary survey value vector U, using known to one group
The measured value of distribution of conductivity U as the reference voltageref, surveyed boundary survey value U to be normalized, after normalization
Boundary survey value vector is
2) energy function based on residual error function and geometric constraints building shape inversion problem:
Wherein, R (x) is residual error item, PΓIt (x) is geometric constraints item, U (x) is the content feature modeling by estimating
And the estimated value of the boundary voltage gone out,Indicate square of 2- norm, symbol ' and " indicate x (s) to the first differential of s and two
Rank differential,Indicating that the curve along object boundary Γ integrates, hyper parameter α and β are used to adjust the degree of geometric constraints,
Distance change on power item constraint boundary between two consecutive points, for controlling the level of stretch on boundary, on stiffness term restrained boundary
Disturbance, for controlling the curvature on boundary；
3) according to variation principle, the Optimal Boundary estimation x for minimizing energy function ε (x) meets following Lagrange
Equation:
Wherein, symbol " " and " indicate x (s) to the quadravalence differential and second-order differential of s；M is the sum of boundary survey value；N is
The exterior normal direction vector on content boundary；JiIndicate measured value UiAbout unit-boundary along exterior normal direction position at point x ∈ Γ
The sensitivity of shifting, its calculation formula is:
Wherein, κ=σk/σbIt is content conductivityσkWith background media conductivityσbRatio；φ is calculated by direct problem
Boundary electric potentials out；It is the differential operator along boundary exterior normal direction；It is the differential operator along boundary tangential direction；Ii
=[I1,I2,…,IL]TIt is given exciting current vector, L is number of poles, MiBe from the measurement that excitation measurement strategies determine to
Amount；
Sliding-model control is carried out to the variable in energy function, by series of discrete point [x (s1),x(s2),…,x(sN)]
Content boundary is characterized, wherein N is the sum of boundary point, and the matrix form of Lagrange's equation is expressed as follows:
Wherein, N is N × N-dimensional diagonal matrix, and diagonal element is [n1n2,…,nN]；J is M × N-dimensional sensitivity matrix；
A is five diagonal band matrix of circulation:
Wherein, a=α/δ s2, b=β/δ s2, c=-a-4b, d=2a+6b；
4) Lagrange's equation is iteratively solved using semi-implicit method；
5) by the sampled point [x (s in successive ignition back boundary estimated value1),x(s2),…,x(sN)] can Step wise approximation mesh
The real border of content is marked, is fitted to obtain the local arc length parameters x (s) for rebuilding content boundary later by discrete point, into
And realize content Boundary Reconstruction.
Detailed description of the invention
Fig. 1 is the sensor structure used in specific embodiment and target contents to be reconstructed；
Fig. 2 is the schematic diagram of the content Boundary Reconstruction method based on geometric constraints；
Fig. 3 is the Boundary Reconstruction result under different conductivity contrasts and different signal-to-noise ratio that emulation obtains；
Fig. 4 be experiment obtain based under pixels approach and Method On Shape to single content rebuild comparing result；
Fig. 5 be experiment obtain based under pixels approach and Method On Shape to double contents rebuild comparing result.
Specific embodiment
The realization step of method involved in the present invention described in detail below, it is intended to it is described as the embodiment of the present invention, and
Non- is the unique forms that the present invention realizes, can realize that the embodiment of identical structure and function also should include of the invention to other
In range.
In a particular embodiment, shown in related EIT system sensor such as Fig. 1 (a), system includes 16 electrodes,
It is even to be distributed in outside tested field domain.To simplify description, it is described in the present embodiment for single content Boundary Reconstruction.In mostly
Inclusion Boundary Reconstruction can directly be extended by the method in the present embodiment.EIT system is using current excitation voltage measurement and excitation electricity
The mode that pole does not measure, the boundary voltage under acquisition cycle motivation circulation measurement on each electrode constitute boundary survey value vector U.
The specific implementation flow of the embodiment mainly includes following steps:
(1) parametrization characterization target contents boundary
As shown in Fig. 1 (b), boundary Γ can be characterized as target contents to be reconstructed by a series of local arc length parameterizeds
X (s), wherein x ∈ Γ indicates that the point set on object boundary, s ∈ [0,1] are local arc length parameters.
The mentioned method of the present invention is a kind of absolute imaging method, in order to reduce this method to model error and noise jamming
Sensibility, a kind of method that the present invention proposes Virtual Calibration.It is used as using the measured value of one group of known conductivity distribution with reference to electricity
Press Uref, surveyed boundary survey value U to be normalized.Boundary survey value vector after normalizationIt may be expressed as:
The purpose of Boundary Reconstruction method is exactly that the boundary of target contents to be reconstructed is estimated by boundary voltage measured value
It is distributed x (s).
(2) energy function based on residual error function and geometric constraints building shape inversion problem
According to variation principle, solves Boundary Reconstruction problem and be equal to minimum residual error function.Different from common energy letter
Number is constituted, present invention combination approximate error, and additionally constructs new energy function using geometric constraints, such as following formula institute
Show:
ε (x)=R (x)+PΓ(x)
Wherein, R (x) is residual error item, PΓIt (x) is geometric constraints item.
Residual error item measures the approximate error between estimated voltage value and measurement voltage value:
WhereinFor boundary voltage measured value, U (x) is the content feature modeling by estimating and the boundary voltage that goes out is estimated
Evaluation.Indicate square of 2- norm.Residual error item is changed by minimizing the difference between estimated voltage value and measurement voltage value
Into the estimation being distributed to target contents boundary.But due to the pathosis of EIT problem, the process be highly prone to measurement noise and
The influence of model error, so as to cause the estimated result of inaccuracy.
In order to improve problem above, the present invention uses geometric constraints as regularization term, for adding to energy function
With constraint.Geometric constraints are illustrated by tension and rigidity:
Wherein, symbol ' and " indicate x (s) to the first differential and second-order differential of s,It indicates along object boundary Γ's
Curve integral, hyper parameter α and β are used to adjust the degree of geometric constraints.On tension item constraint boundary between two consecutive points away from
From variation, it is used to control the level of stretch on boundary.Disturbance on stiffness term restrained boundary, it is used to control the curvature on boundary.
(3) sliding-model control is carried out to the variable in energy function
Using shown in the present invention constrain in the case where, minimize energy function ε (x) be exactly find completely fitting measured value and
Keep the optimal compromise between stability boundaris.According to variation principle, the Optimal Boundary for minimizing energy function ε (x) estimates x
Meet following Lagrange's equation:
Wherein, symbol " " and " indicate x (s) to the quadravalence differential and second-order differential of s；M is the sum of boundary survey value；N is
The exterior normal direction vector on content boundary；JiIndicate measured value UiAbout unit-boundary along exterior normal direction position at point x ∈ Γ
The sensitivity of shifting, its calculation formula is:
Wherein, κ=σk/σbIt is content conductivityσkWith background media conductivityσbRatio；φ is calculated by direct problem
Boundary electric potentials out；It is the differential operator along boundary exterior normal direction；It is the differential operator along boundary tangential direction；Ii
=[I1,I2,…,IL]TIt is given exciting current vector, L is number of poles, MiBe from the measurement that excitation measurement strategies determine to
Amount.
To solve the above Lagrange's equation, sliding-model control need to be carried out to variable.Using a constant interval δ s couple
Local arc length parameters s is sampled, then content boundary to be reconstructed can be by series of discrete point [x (s1),x(s2),…,x(sN)]
It indicates, wherein N is the sum of boundary point.In view of in above-mentioned Lagrange's equation each vector be it is independent and separable, can will
Each vector is separated into the part x and y, uses ujIndicate x (sj) or y (sj), use njIndicate n (sj) x-component or y-component.Lagrange
The matrix form of equation is expressed as follows:
Wherein, N is N × N-dimensional diagonal matrix, and diagonal element is [n1n2,…,nN]；J is M × N-dimensional sensitivity matrix；
A is five diagonal band matrix of circulation:
Wherein, a=α/δ s2, b=β/δ s2, c=-a-4b, d=2a+6b.In above-mentioned matrix equation, first item is by residual error
Item R export, Section 2 is by geometric constraints item PΓExport.Sensitivity matrix J characterizes EIT measured value and content boundary
The local linear relationship of variation.To prevent influence of the truncated error generated in calculating process to residual error item R and gradient r, at this
Scaling processing is carried out to sensitivity matrix in embodiment, the sensitivity matrix after scaling is equal to J/max (J).
(4) Lagrange's equation is iteratively solved
The present invention iteratively solves the above Lagrange's equation using semi-implicit method.First, it is assumed that on the right of equation
Constant and the equation left side item changes with iterative steps when item.Secondly, it is assumed that r is constant and is equal in iterative process each time
In rt-1.Finally, it is assumed that matrix A is known in time t.
Such as lower boundary EVOLUTION EQUATION can be obtained according to above-mentioned hypothesis:
Wherein, δ t is time step.Thus the iteration form of the mentioned Boundary Reconstruction method of the present invention is derived:
Wherein S=δ tA+E is the filter by being calculated in geometrical constraint item, and E is unit matrix.
Fig. 2 illustrates the Computing Principle of the mentioned method of the present invention.In iterative process each time, the mentioned boundary weight of the present invention
It builds the solver first step and optimum displacement gradient r is calculated by gradient descent method minimum residual error function Rt-1, second step later
X is estimated using smoothing filter S adjustment boundary immediatelyt-1+δtrt-1.In an iterative process, hyper parameter α and β and sampling step length δ
S is invariable.The selection of time step δ t need to comprehensively consider stability and convergence rate, rule of thumb, to realize light
The selection of sliding and stable boundary approximate procedure, time step need to meet δ s2< δ tmax (Nr) < δ s.Iterative process is in residual error
Sufficiently small or stopping when being approximately constant, therefore, stop condition setting are as follows:
Rt≤τ1or|Rt-Rt-1|/Rt-1≤τ2
Wherein, τ1It is the tolerance limit of residual error；τ2The tolerance of residual error variable quantity limits.Tolerance limit is selected according to trial-and-error method.
(5) content boundary profile is fitted
According to the above method, by the sampled point [x (s in successive ignition back boundary estimated value1),x(s2),…,x(sN)]
Can Step wise approximation target contents real border, later by discrete point fitting can be realized with high stability and high-precision
Content Boundary Reconstruction.
Result of implementation: test is emulated and tested to the embodiment above.Fig. 3 be emulation obtain in different conductivity
Boundary Reconstruction under contrast and different signal-to-noise ratio as a result, Comparative result range be the conductivity contrast of 1.25-1000 with
And joined 60dB, the Boundary Reconstruction result under 40dB and 20dB signal-to-noise ratio noise and noiseless, it can be seen that although rebuilding knot
Fruit is deteriorated with the increase of noise, but the mentioned method of the present invention still has good noise immunity, under 20dB noise, is mentioned
Method still can preferably reconstruct the position of target contents, size and shape.Meanwhile the noise immunity of mentioned method can lead to
Raising conductivity contrast is crossed to improve.Fig. 4 be experiment obtain based under pixels approach and Method On Shape to single content weight
The comparing result built, Comparative result NOSER algorithm imaging results, sparse imaging results based on L1 norm, TV regularization are calculated
Method imaging results, and the imaging results of the Boundary Reconstruction method (abbreviation GCBR) based on geometric constraints proposed.It can
To find out, the shape for the reconstruction content that method pixel-based can only be rough, wherein in NOSER algorithm can hardly identify
The different shape of inclusion, and L1 and TV regularization can only substantially distinguish oval and triangle content, cannot distinguish between the rectangular and heart
The content of shape.And mentioned method accurate can then reconstruct the geometry of target contents.Fig. 5 experiment obtains
The comparing result that double contents are rebuild, it can be seen that compared to image rebuilding method pixel-based, the side GCBR proposed
Method has better reconstructed results to non-circular content.
Claims (1)
1. a kind of electrical impedance tomography content Boundary Reconstruction method based on geometric constraints, including the following steps:
1) using the boundary of part arc length parameters x (s) characterization target contents, wherein x ∈ Γ indicates the point set on object boundary,
S ∈ [0,1] is local arc length parameters, constructs sensitivity matrix J and obtains boundary survey value vector U, using conductance known to one group
The measured value U as the reference voltage of rate distributionref, surveyed boundary survey value U to be normalized, the boundary after normalization
Measured value vector is
2) energy function based on residual error function and geometric constraints building shape inversion problem:
Wherein, R (x) is residual error item, PΓIt (x) is geometric constraints item, U (x) is the content feature modeling by estimating and goes out
Boundary voltage estimated value,Indicate square of 2- norm, symbol ' and " indicate that x (s) is micro- to the first differential and second order of s
Point, ∮ΓDs indicates that the curve along object boundary Γ integrates, and hyper parameter α and β are used to adjust the degree of geometric constraints, tension
Distance change on item constraint boundary between two consecutive points, for controlling the level of stretch on boundary, disturbing on stiffness term restrained boundary
It is dynamic, for controlling the curvature on boundary；
3) according to variation principle, the Optimal Boundary estimation x for minimizing energy function ε (x) meets following Lagrange's equation:
Wherein, symbol " " and " indicate x (s) to the quadravalence differential and second-order differential of s；M is the sum of boundary survey value；N is to include
The exterior normal direction vector on object boundary；JiIndicate measured value UiAbout unit-boundary along the displacement of exterior normal direction at point x ∈ Γ
Sensitivity, its calculation formula is:
Wherein, κ=σk/σbIt is content conductivityσkWith background media conductivityσbRatio；φ is calculated by direct problem
Boundary electric potentials；▽nIt is the differential operator along boundary exterior normal direction；▽tIt is the differential operator along boundary tangential direction；Ii=
[I1,I2,…,IL]TIt is given exciting current vector, L is number of poles, MiBe from the measurement that excitation measurement strategies determine to
Amount；
Sliding-model control is carried out to the variable in energy function, by series of discrete point [x (s1),x(s2),…,x(sN)] characterization
Content boundary, wherein N is the sum of boundary point, and the matrix form of Lagrange's equation is expressed as follows:
Wherein, N is N × N-dimensional diagonal matrix, and diagonal element is [n1n2,…,nN]；J is M × N-dimensional sensitivity matrix；A is
Recycle five diagonal band matrix:
Wherein, a=α/δ s2, b=β/δ s2, c=-a-4b, d=2a+6b；
4) Lagrange's equation is iteratively solved using semi-implicit method；
5) by the sampled point [x (s in successive ignition back boundary estimated value1),x(s2),…,x(sN)] can be in Step wise approximation target
The real border of inclusion is fitted to obtain the local arc length parameters x (s) for rebuilding content boundary, Jin Ershi later by discrete point
Existing content Boundary Reconstruction.
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN109946388A (en) * | 2019-02-20 | 2019-06-28 | 天津大学 | Based on electricity/ultrasonic double-mode state content Boundary Reconstruction method that statistics is inverse |
| CN109946388B (en) * | 2019-02-20 | 2021-04-27 | 天津大学 | Electrical/ultrasonic bimodal inclusion boundary reconstruction method based on statistical inversion |
| CN110827928A (en) * | 2019-11-04 | 2020-02-21 | 曲阜师范大学 | Method for constructing single-layer carbon fiber reinforced plastic conductivity model based on boundary element method |
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